Judy Kennedy

Basins of Wada

Abstract We describe situations in which there are several regions (more than two) with the Wada property, namely that each point that is on the boundary of one region is on the boundary of all. We argue that such situations arise even in studies of the forced damped pendulum, where it is possible to have three attractor regions coexisting, and the three basins of attraction have the Wada property.

Orbits of the pseudocircle

The following theorem is proved. THEOREM. The pseudocircle has uncountably many orbits under the action of its homeomorphism group. Each orbit is the union of uncountably many composants. A pseudocircle is a circularly chainable, hereditarily indecomposable, separating plane continuum. R. H. Bing [1] constructed a pseudocircle in 1951 and asked two questions. Are any two pseudocircles homeomorphic? Is a pseudocircle homogeneous? In 1968, L. Fearnley [6] proved that the answer to the first question is yes, and Fearnley [5] and J. T. Rogers, Jr. [14] independently proved that the answer to the second question is no. Since the advent of the Effros theorem, several elegant proofs of the nonhomogeneity of the pseudocircle have appeared [7, 10, and 13]. In 1968, after seeing the proof that the pseudocircle is not homogeneous, F. B. Jones asked the second author how many orbits the pseudocircle had under the action of its homeomorphism group, but the matter was not pursued further. Recently, however, the question has arisen again, and Wayne Lewis [3] has asked if the pseudocircle has uncountably many orbits. The purpose of this paper is to prove the following theorem, which gives an affirmative answer to this question. THEOREM. The pseudocircle has uncountably many orbits under the action of its homeomorphism group. Each of these orbits is the union of uncountably many composants. An interesting sidelight of the proof is the construction of two homeomorphic sets, one being an open set of the pseudocircle and the other being an open set of the pseudoarc. During the proof we construct an uncountable, abelian subgroup of the homeomorphism group of the pseudocircle. The homeomorphism constructed by Handel [8] might be an element of this group. Each orbit of the pseudocircle is a Borel set [4]. Lewis has announced that no orbit can be a G6. In the course of our argument we prove this, but we do not determine further restrictions on the type of the Borel set for the orbit. Question. What classes of Borel sets occur as the orbits of the pseudocircle? In particular, are each two orbits of the same class? Received by the editors May 1, 1984 and, in revised form, August 13, 1985. 1980 Mathematics Subject Classification. Primary 54F20, 54F50.

Bizarre topology is natural in dynamical systems

We describe an example of a

The theory of chaotic attractors

C^\infty

The topology of stirred fluids

diffeomorphism on a 7--manifold which has a compact invariant set such that uncountably many of its connected components are pseudocircles. (Any 7--manifold will suffice.) Furthermore, any diffeomorphism which is sufficiently close (in the

Indecomposable continua in dynamical systems with noise: Fluid flow past an array of cylinders

C^1

Pseudocircles in dynamical systems

metric) to the constructed map has a similar invariant set, and the dynamics of the map on the invariant set are chaotic.

The topology of fluid flow past a sequence of cylinders

Contents: Preface.-Introduction.- E.N. Lorenz, Deterministic nonperiodic flow.- K. Krzyzewski and W. Szlenk, On invariant measures for expanding differentiable mappings.- A. Lasota and J.A. Yorke, On the existence of invariant measures for piecewise monotonic transformations.- R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows.- T.-Y. Li and J.A. Yorke, Period three implies chaos.- R.M. May, Simple mathematical models with very complicated dynamics.- M. Henon, A two- dimensional mapping with a strange attractor.- E. Ott, Strange attractors and chaotic motions of dynamical systems.- F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations.- D. J. Farmer, E. Ott and J.A. Yorke, The dimension of chaotic attractors .- P. Grassberger and I. Procaccia, Measuring the strangeness of strange attractors.- M. Rychlik, Invariant measures and variational principle for Lozi applications.- P. Collet and Y. Levy, Ergodic properties of the Lozi mappings .- J. Milnor, On the Concept of Attractor.-L.-S. Young, Bowen-Ruelle Measures for certain Piewise Hyperbolic Maps.-J.-P. Eckmann and D. Ruelle, Ergodic Theory of Chaos and Strange Attractors.-M.R. Rychlik, Another Proof of Jakobson Theorem and Related Results.- C. Grebogi, E. Ott, and J.A. Yorke, Unstable periodic Orbits and the Dimensions of Multifractal Chaotic Attractors.-P. Gora and A. Boyarsky, Absolutely Continuous Invariant Measures for Piecewise Expanding C Transformation in R.-M. Benedicks and L.-S. Young, Sinai-Bowen-Ruelle Measures for Certain Henon Maps.-M. Dellnitz and O. Junge, On the Approximation of Complicated Dynamical Behavior.-M. Tsujii, Absolutely Continuous Invariant Measures for Piecewise Real-Analytic Expanding Maps on the Plane.-J.F. Alves, C.Bonatti, and M. Viana, SRB Measures for Partially Hyperbolic Systems Whose Central Direction is Mostly Expanding.- B.R. Hunt, J.A. Kennedy, T.-Y. Li, and H.E. Nusse, SLYRB Measures: Natural Invariant Measures for Chaotic Systems.- Credits

How Indecomposable Continua Arise in Dynamical Systemsa

Abstract There are simple idealized mathematical models representing the stirring of fluids. The models we consider involve two fluids entering a chamber, with the overflow leaving it. The stirring creates a Cantor-like, but connected, boundary between the fluids that is best described point-set topologically. We prove that in many cases the boundary between the fluids is an indecomposable continuum.

SLYRB measures: natural invariant measures for chaotic systems

Standard dynamical systems theory is based on the study of invariant sets. However, when noise is added, there are no bounded invariant sets. Our goal is then to study the fractal structure that exists even with noise. The problem we investigate is fluid flow past an array of cylinders. We study a parameter range for which there is a periodic oscillation of the fluid, represented by vortices being shed past each cylinder. Since the motion is periodic in time, we can study a time-1 Poincare map. Then we add a small amount of noise, so that on each iteration the Poincare map is perturbed smoothly, but differently for each time cycle. Fix an x coordinate x(0) and an initial time t(0). We discuss when the set of initial points at a time t(0) whose trajectory (x(t),y(t)) is semibounded (i.e., x(t)>x(0) for all time) has a fractal structure called an indecomposable continuum. We believe that the indecomposable continuum will become a fundamental object in the study of dynamical systems with noise. (c) 1997 American Institute of Physics.